Disentangling geometric and dissipative origins of negative Casimir entropies

Dissipative electromagnetic response and scattering geometry are potential sources for the appearance of a negative Casimir entropy. We show that the dissipative contribution familiar from the plane-plane geometry appears also in the plane-sphere and the sphere-sphere geometries and adds to the negative Casimir entropy known to exist in these geometries even for perfectly reflecting objects. Taking the sphere-sphere geometry as an example, we carry out a scattering-channel analysis, which allows us to distinguish between the contributions of different polarizations. We demonstrate that dissipation and geometry share a common feature making possible negative values of the Casimir entropy. In both cases there exists a scattering channel whose contribution to the Casimir free energy vanishes in the high-temperature limit. While the mode-mixing channel is associated with the geometric origin, the transverse electric channel is associated with the dissipative origin of the negative Casimir entropy. By going beyond the Rayleigh limit, we find even for large distances that negative Casimir entropies can occur also for Drude-type metals provided the dissipation strength is sufficiently small.

Figure 1

The Casimir entropy $\mathcal{S}$ scaled by its high-temperature value ${\mathcal{S}}_{\text{HT}}^{\text{P}}$ for perfect conductors is shown as a function of temperature for the (a) plane-plane geometry, (b) sphere-sphere geometry, and (c) plane-sphere geometry. While the dashed lines represent the entropy for perfectly conducting objects, the solid lines correspond to Drude-type conductors with the damping constant $\gamma d/c={10}^{-2},{10}^2,$ and ${10}^4$ increasing from bottom to top and the plasma frequency given by ${\omega }_{P}d/2\pi c=400$. The sphere radii are chosen as $R=d/20$. The arrows indicate the effect of the missing zero-frequency contribution to the Casimir entropy for Drude-type metals with respect to perfect conductors.

Figure 2

The ratios of Mie coefficients $-b_1^{\text{D}}/a_1^{\text{D}}$ and $a_2^{\text{D}}/a_1^{\text{D}}$ as a function of the wave number $k$ are shown for a Drude-type metal sphere with radius $R$. The plasma frequency and the damping constant are given by ${\omega }_{P}R/2\pi c=20$ and $\gamma /{\omega }_P={10}^{-4}$. Three different regimes can be distinguished. In the Rayleigh regime, TE scattering is negligible, while in the dipole regime, TE and TM scattering are of comparable strength. In the multipole regime, the scattering of higher multipole waves becomes relevant. The dashed lines indicate the approximations valid in the first two regimes with references to the corresponding equations.

Figure 3

The contributions from (a) the TM channel, (b) the TE channel, and (c) the polarization-mixing channels to the Casimir entropy are shown as function of temperature for a sphere-sphere geometry with $d/R=20$. All contributions are scaled by the total high-temperature Casimir entropy ${\mathcal{S}}_{\text{HT}}^{\text{P}}$ for perfectly conducting spheres. The parameters are chosen as in Fig. 1, i.e., the dashed lines correspond to perfect metal spheres while the solid black and red lines correspond to Drude-type metal spheres with $\gamma d/c={10}^2$ and ${10}^4$, respectively, and ${\omega }_{P}d/2\pi c=400$. For the sake of clarity, panel (a) displays only one line because all other lines deviate by not more than one line width. To facilitate the comparison between the contributions of the three types of scattering channels, the scales on the vertical axes are chosen to be equal.

Figure 4

The contributions of the different scattering channels to the scaled free energy $\tilde{\mathcal{F}}={(d/R)}^6\mathcal{F}$ are shown as a function of temperature. The distance between the two Drude-type metal spheres is given by $d/R=20$ and the material parameters are $\gamma d/c=10$ and ${\omega }_{P}d/2\pi c=400$. The inset displays the low-temperature behavior of the TE channel and demonstrates that the corresponding contribution to the Casimir entropy vanishes in the zero-temperature limit.

Figure 5

The Casimir entropy $\mathcal{S}$ is presented as a function of the distance $d$ between two perfectly conducting spheres of radius $R$ at $2\pi k_{B}TR/\hslash c=1$ for different levels of approximation. Results obtained within the single round-trip approximation are displayed as white and gray points. While the first ones correspond to the dipole approximation, the latter include higher multipoles up to ${\ell }_{\text{max}}=60$. The numerically exact result depicted by black points is obtained on the basis of Eq. (2) with ${\ell }_{\text{max}}=60$. The inset shows the relative difference of the dipole approximation ${\mathcal{S}}^{\mathrm{dip}}$ and the exact result ${\mathcal{S}}^{\mathrm{ex}}$.

Figure 6

The scaled Casimir entropy $\tilde{\mathcal{S}}={(d/R)}^6\mathcal{S}$ as a function of temperature is shown as solid line accounting for multipoles up to ${\ell }_{\text{max}}=15$. The contributions arising from dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole scattering are depicted as dashed, dash-dotted, and dotted line, respectively. The distance between the perfectly conducting spheres is $d=2.75R$.

Figure 7

The Casimir entropy $\mathcal{S}$ is presented as a function of the distance $d$ between Drude-type metal spheres of radius $R$ at $2\pi k_{B}TR/\hslash c=1$ within the single round-trip approximation. The material parameters correspond to those of the best conductor represented by a blue curve in Fig. 1, i.e., ${\omega }_{P}R/2\pi c=20$ and ${\omega }_P/2\pi \gamma =4\times {10}^4$. The white points correspond to the dipole approximation while the gray points include higher multipoles up to ${\ell }_{\text{max}}=60$. Numerically exact results accounting for multiple round-trips cannot be distinguished from the gray points in this figure and are therefore not shown. The inset displays the relative difference of the dipole approximation ${\mathcal{S}}^{\mathrm{dip}}$ and the exact result ${\mathcal{S}}^{\mathrm{ex}}$.